Let \(T_*\)  denote the total complex of the double complex \(\lbrace  C_{p,q}, \delta _{p,q}, \partial _{p,q} \rbrace _ {p,q \in \mathbb {Z}}\) . The Mayer-Vietoris spectral sequence of the collection \(\lbrace  X^j \rbrace _ {j\in J}\)  is defined as the spectral sequence associated with the first filtration \(F^{\text{I}}_p\)  of \(T_*\) . Using the exact sequence (REF ) and the second filtration \(F^{\text{II}}_p\)  of \(T_*\)  it is not hard to prove
that the homology of the total complex coincides with the homology of the union of the collection of subcomplexes, that is \(H_*(T_*) \cong H_* (\cup _{j\in J} X^j)\) . This implies that the Mayer-Vietoris spectral sequence converges to \(H_* (\cup _{j\in J} X^j)\) .
Let us state
a more general result that implies this fact, as we need it also to prove Theorem REF  in Section . This result is stated for example in [1]}, and a proof can be found, in a cohomological setting, in [2]}.
